Variational Iteration Method for Solving Telegraph Equations

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1 Availabl a hp://pvam.d/aam Appl. Appl. Mah. ISSN: 9-9 Vol. I (J 9) pp. (Prvioly Vol. No. ) Applicaio ad Applid Mahmaic: A Iraioal Joral (AAM) Variaioal Iraio Mhod for Solvig Tlgraph Eqaio Syd Taf Mohyd-Di ydaf@homail.com Mhammad Alam Noor oormalam@homail.com Khalida Iaya Noor khalidaoor@homail.com Dparm of Mahmaic COMSATS Ii of Iformaio Tchology Ilamabad Pakia Rcivd: Dcmbr 8; accpd: Fbrary 9 Abrac I hi papr w apply h variaioal iraio mhod (VIM) for olvig lgraph qaio which ari i h propagaio of lcrical igal alog a lgraph li. Th ggd algorihm i mor ffici ad air o hadl a compar o h dcompoiio mhod. Nmrical rl how h fficicy ad accracy of h propod VIM. Kyword: Variaioal Iraio Mhod; Lagrag Mliplir; Tlgraph Eqaio MSC () No.: N. Irodcio Th lgraph qaio appar i h propagaio of lcrical igal alog a lgraph li digial imag procig lcommicaio igal ad ym [ Abdo ad Solima () Wazwaz ( )]. Th adard form of h lgraph qaio i giv a

2 AAM: Ir. J. Vol. I (J 9) [Prvioly Vol. No. ] a b c whr a b c ar coa rlad o riac idcac capaciac ad codcac of h cabl. Th baic moivaio of hi papr i h applicaio of variaioal iraio mhod (VIM) for olvig lgraph qaio. Th VIM wa dvlopd ad formlad by H for olvig vario phyical problm [ H (999 8)]. Th mhod ha b rmly fl for divrifid iiial ad bodary val problm ad ha h poial o cop wih h vrailiy of h compl ar of phyical problm [ Abbabady () Abdo ad Solima () H (999 8) Noor ad Mohyd-Di ( 8) Mohyd-Di al. (8 9) X ()]. A wid cla of iiial ad bodary val problm icldig Riccai diffrial qaio ady flow hrogh a poro mdim Brgr ad copld Brgr qaio oliar ocillaor highr-ordr bodary val problm of vario ordr pag flow aoomo ordiary diffrial ym Fihr qaio Evolio qaio diffio qaio iglar problm KdV parabolic qaio igro-diffrial qaio chmiry problm Boiq qaio Schrödigr qaio Hlmholz qaio Si-Gordo qaio ha b ackld ccflly i accordac wih hir phyical ar by h propod variaioal iraio mhod (VIM) [ Abbabady () Abdo ad Solima () H (999 8) Noor ad Mohyd-Di ( 8) Mohyd-Di al. (8 9) X ()]. I i o b highlighd ha h of Lagrag mliplir i variaioal iraio mhod (VIM) rdc h cciv applicaio of h igral opraor miimiz h compaioal work o a agibl lvl whil ill maiaiig a vry high lvl of accracy ad hc i a clar advaag of hi chiq ovr h dcompoiio mhod. Th VIM i alo idpd of h mall paramr ampio (which i ihr o hr i h phyical problm or difficl o loca) ad hc i mor covi o apply a compar o h radiioal prrbaio mhod. I i worh mioig ha h VIM i applid wiho ay dicrizaio rriciv ampio or raformaio ad i fr from rod off rror. W apply h propod VIM for all h oliar rm i h problm wiho dicrizig ihr by fii diffrc or pli chiq a h od ivolv laborio calclaio copld wih a rog poibiliy of h ill-codiiod rla qaio which i a complicad problm o olv. Morovr lik h mhod of paraio of variabl ha rqir iiial ad bodary codiio h VIM provid h olio by ig h iiial codiio oly [ Abbabady () Abdo ad Solima () H (999 8) Noor ad Mohyd-Di ( 8) Mohyd-Di al. (8 9) X ()]. Th propod variaioal iraio mhod (VIM) ca b applid o a mbr of phyical problm rlad o flid mchaic icldig bodary layr flow wih poial or algbraic propri Vo Karma wirlig vico flow oliar progriv wav i dp war poro mdim fiacial mahmaic dp hallow war wav lcrical igal alog a lgraph li digial imag procig lcommicaio igal ad ym bam dflcio hory qam fild hory rlaiviic phyic dipriv wav-phoma plama

3 Mohyd-Di l al. phyic arophyic oliar opic girig ad applid cic [ Mohyd-Di al. (9)]. Nmrical rl how h compl rliabiliy of h propod chiq.. Variaioal Iraio Mhod (VIM) To illra h baic cocp of h H VIM w coidr h followig gral diffrial qaio L N g() () whr L i a liar opraor N a oliar opraor ad g() i h ihomogo rm. Accordig o variaioal iraio mhod [ Abbabady () Abdo ad Solima () H (999 8) Noor ad Mohyd-Di ( 8) Mohyd-Di al. (8 9) X ()] w ca corc a corrcio fcioal a follow ( ) ( ) ( L ( ) N ( ) g( )) d () whr i a Lagrag mliplir [ H (999 8)] which ca b idifid opimally via variaioal iraio mhod. Th bcrip do h h approimaio i coidrd a a rricd variaio. i.. ; () i calld a corrcio fcioal. Th olio of h liar problm ca b olvd i a igl iraio p d o h ac idificaio of h Lagrag mliplir. Th pricipl of variaioal iraio mhod ad i applicabiliy for vario kid of diffrial qaio ar giv i [ H (999 8)]. I hi mhod i i rqird fir o drmi h Lagrag mliplir opimally. Th cciv approimaio of h olio will b radily obaid po ig h drmid Lagrag mliplir ad ay lciv fcio coqly h olio i giv by lim.. Nmrical Applicaio I hi cio w apply h variaioal iraio mhod (VIM) for olvig lgraph qaio. Nmrical rl ar vry coragig. Eampl.. Coidr h followig lgraph qaio

4 AAM: Ir. J. Vol. I (J 9) [Prvioly Vol. No. ] wih bodary codiio ) ( ) ( ad h iiial codiio. Th corrcio fcioal i giv by. d Makig h corrcio fcioal aioary h Lagrag mliplir ca b idifid a w obai h followig iraiv formla. d Coqly followig approima ar obaid!!!!!!!!!!! 9! 8!!!!!! 9 8. Th ri olio i giv by 9! 8!!!!!! 9 8 ad h clod form olio i giv a which i h ac olio.

5 8 Mohyd-Di l al. Eampl.. Coidr h followig lgraph qaio wih bodary codiio ( ) ( ) ad h iiial codiio. Th corrcio fcioal i giv by. d Makig h corrcio fcioal aioary h Lagrag mliplir ca b idifid a w obai h followig iraiv formla. d Coqly followig approima ar obaid!!!!!!!!!!!. Th ri olio i giv by!!!!! ad h clod form olio i giv a

6 AAM: Ir. J. Vol. I (J 9) [Prvioly Vol. No. ] 9 which i h ac olio. Eampl.. Coidr h followig lgraph qaio wih bodary codiio ) ( ) ( ad h iiial codiio. Th corrcio fcioal i giv by. d Makig h corrcio fcioal aioary h Lagrag mliplir ca b idifid a w obai h followig iraiv formla. d Coqly followig approima ar obaid. Procdig a bfor h ac olio i giv a.

7 Mohyd-Di l al.. Coclio I hi papr w applid variaioal iraio mhod (VIM) for olvig lgraph qaio. Th mhod i applid i a dirc way wiho ig liarizaio raformaio prrbaio dicrizaio or rriciv ampio. Th fac ha h propod VIM olv oliar problm wiho ig Adomia polyomial i a clar advaag of hi algorihm ovr h dcompoiio mhod. Ackowldgm Th ahor ar highly grafl o boh h rfr ad Prof. Dr. A. M. Haghighi for hir vry corciv comm. W wold lik o hak Dr. S. M. Jaid Zaidi Rcor CIIT for h proviio of cll rarch facilii ad virom. REFERENCES Abbabady S. (). Nmrical olio of oliar Kli-Gordo qaio by variaioal iraio mhod. Ira. J. Nmr. Mh. Egrg. pp Abdo M. A. ad Solima A. A. (). Variaioal iraio mhod for olvig Brgr ad copld Brgr qaio. J. Comp. Appl. Mah. 8 pp. -. Abdo M. A. ad Solima A. A. (). Nw applicaio of variaioal iraio mhod. Phy. D (-) pp () -8. H J. H. (999). Variaioal iraio mhod A kid of o-liar aalyical chiq om ampl. Ira. J. Noli. Mch. () pp H J. H. (). Variaioal iraio mhod for aoomo ordiary diffrial ym. Comp. Mah. Appl. (-) pp. -. H J. H. ad W X. H (). Corcio of oliary olio ad compacio-lik olio by variaioal iraio mhod. Chao Solio & Fracal 9 () pp. 8-. H J. H. (). Variaioal iraio mhod om rc rl ad w irpraio. J. Comp. Appl. Mah. pp. -. H J. H. ad W X. H. (). Variaioal iraio mhod: Nw dvlopm ad applicaio. Comp. Mah. Appl. pp H J. H. (). Th variaioal iraio mhod for ighh-ordr iiial bodary val problm Phy. Scr. () pp H J. H. (8). A lmary irodcio of rcly dvlopd aympoic mhod ad aomchaic i il girig. I. J. Mod. Phy. B () pp H J. H. (). Som aympoic mhod for rogly oliar qaio. I. J. Mod. Phy. () pp Mohyd-Di S. T. Noor M. A. ad Noor K. I. (9). Travllig wav olio of vhordr gralizd KdV qaio by variaioal iraio mhod ig Adomia polyomial. I. J. Mod. Phy. B. Mohyd-Di S. T. Noor M. A. ad Noor K. I. (9). Modifid variaioal iraio mhod for olvig volio qaio. I. J. Mod. Phy. B. Mohyd-Di S. T. Noor M. A. ad Noor K. I. (9). Solio of iglar qaio by H

8 AAM: Ir. J. Vol. I (J 9) [Prvioly Vol. No. ] variaioal iraio mhod. I. J. Noli. Sci. Nm. Sim. -. Mohyd-Di S. T. Noor M. A. ad Noor K. I. (9). O h coplig of polyomial wih corrcio fcioal. I. J. Mod. Phy. B. Noor M. A. ad Mohyd-Di S. T. (). Variaioal iraio chiq for olvig highr ordr bodary val problm. Appl. Mah. Comp. 89 pp Noor M. A. ad Mohyd-Di S. T. (). A ffici mhod for forh ordr bodary val problm Comp. Mah. Appl. pp. -. Noor M. A. ad Mohyd-Di S. T. (8). Variaioal iraio mhod for olvig highrordr oliar bodary val problm ig H polyomial. I. J. Noli. Sci. Nm. Siml. 9 () pp. -. Noor M. A. ad Mohyd-Di S. T. (8). Variaioal iraio mhod for olvig fifh-ordr bodary val problm ig H polyomial. Mah. Prob. Egg. 8 Aricl ID 99 doi: :/8/99. Noor M. A. ad Mohyd-Di S. T. (8). Modifid variaioal iraio mhod for ha ad wav-lik qaio. Aca Applda. Mahmc. DOI:./ Noor M. A. ad Mohyd-Di S. T. (8). Modifid variaioal iraio mhod for olvig forh-ordr bodary val problm J. Appl. Mah. Compg. Aricl ID: 9 DOI:./9-8-9-z. Noor M. A. ad Mohyd-Di S. T. (8). Variaioal homoopy prrbaio mhod for olvig highr dimioal iiial bodary val problm. Mah. Prob. Egg. 8 Aricl ID 9 doi:./8/9. Noor M. A. ad Mohyd-Di S. T. (9). Variaioal iraio mhod for ady flow of ga hrogh a poro mdim ig H polyomial ad Pad approima. Comp. Mah. Appl. Noor M. A. ad Mohyd-Di S. T. (9) Modifid variaioal iraio mhod for olvig Hlmholz qaio. Comp. Mah. Modlg. Noor M. A. ad Mohyd-Di S. T. (9). Variaioal iraio mhod for olvig wlfhordr bodary val problm ig H polyomial. Comp. Mah. Modlg. Noor M. A. ad Mohyd-Di S. T. (8). Solio of iglar ad oiglar iiial ad bodary val problm by modifid variaioal iraio mhod. Mah. Prob. Egg. 8 Aricl ID 9 pag doi:./8/9. Wazwaz A. M. (). Th modifid dcompoiio mhod for aalyic ram of diffrial qaio. Appl. Mah. Comp. Pp. -. Wazwaz A. M. (). Parial diffrial qaio mhod ad applicaio. A. A. Balkma Pblihr. X L. (). Th variaioal iraio mhod for forh-ordr bodary val problm. Chao Solio & Fracal i pr.

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